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I nte. J. Ma. Math. Vo1. {1978)1-12 BEHAVIOR OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS RINA LING Department of Mathematics California State University Los Angeles, California 90032 (Received November 29, 1977 and in revised form March 31, 1978) ABSTRACT. Qualitative behavior of second order nonlinear differential equations with variable coefficients is studied. It includes properties such as posltlvlty, number of zeroes, oscillatory behavior, boundedness and monotoniclty of the solutions. i. INTRODUCTION. Second order nonlinear differential equations of the form y(t) + p(t)y(t) + q(t)yn(t) 0 where n is an integer >_ 2, occur in many physical problems, such as the mass- spring systems and satellite (see Ames [i], Mclachlan [2] and Struble [3]) and nuclear energy distribution (see Canosa and Cole [4, 5]). In this work, qualitative behavior of real-valued solutions of (i.i) is studied. With certain conditions on the coefficients p(t) and q(t), and n, properties such as posltivlty, number of zeroes, boundedness and monotonlclty

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Page 1: Behavior of SON Differential Equations

Inte. J. Ma. Math.Vo1. {1978)1-12

BEHAVIOR OF SECOND ORDER NONLINEARDIFFERENTIAL EQUATIONS

RINA LINGDepartment of MathematicsCalifornia State University

Los Angeles, California 90032

(Received November 29, 1977 and in revised form March 31, 1978)

ABSTRACT. Qualitative behavior of second order nonlinear differential equations

with variable coefficients is studied. It includes properties such as posltlvlty,

number of zeroes, oscillatory behavior, boundedness and monotoniclty of the

solutions.

i. INTRODUCTION.

Second order nonlinear differential equations of the form

y(t) + p(t)y(t) + q(t)yn(t) 0

where n is an integer >_ 2, occur in many physical problems, such as the mass-

spring systems and satellite (see Ames [i], Mclachlan [2] and Struble [3]) and

nuclear energy distribution (see Canosa and Cole [4, 5]).

In this work, qualitative behavior of real-valued solutions of (i.i) is

studied. With certain conditions on the coefficients p(t) and q(t), and n,

properties such as posltivlty, number of zeroes, boundedness and monotonlclty

Page 2: Behavior of SON Differential Equations

2 RINA LING

are obtained. It would be assumed that the coefficients and their derivatives

are continuous real-valued functions on the interval of interest. The work is

divided into four parts; the first part deals with the case of p(t) < 0 and

q(t) < 0, the second part with the case of p(t) < 0 and q(t) > O, the third

part with p(t) > 0 and q(t) < 0 and the fourth part with p(t) > 0 and q(t) > O.

Papers in the past, Skldmore [6], Abramovlch [7], Rankln [8], and Grimmer and

Patula [9] have studied behavior of second order linear differential equations.

Nonllnear differential equations have been investigated in Chen [i0] and Chen,

Yeh and Yu [11], the former is on oscillatory behavior of bounded solutlons and

the latter on asymptotic behavior of solutions. The results here are of a

different nature and are independent of theirs.

2. CASE OF p(t) < 0 AND q(t) < 0.

THEOREM 2.1. If (i) p(t) < 0, t > 0, (2) q(t) < O, t > 0 and (3) n

is odd, then either y(t) > O, t > 0 or y(t) < 0, t > 0. The graph is concave

upward for y > 0, and concave downward for y < 0.

PROOF. Equation (i.i) can be written in the form

+ (p(t) + q(t)yn-l)y O. (2.1)

Let z(t) be real-valued and satisfy the linear equation

+ p(t)z 0. (2.2)

By a theorem in Hartman [12, p. 346-347], z(t) has no zero. Since n i is

even and q(t) < 0, we have

p(t) + q(t)yn-1 < p(t)

and by Sturm First Comparison Theorem, z(t) has at least a zero on (0,=), if y

has a zero on (0,), a contradiction.

The concavity of the graph follows from (I.i) written in the form

Page 3: Behavior of SON Differential Equations

NONLINEAR DIFFERENTIAL EQUATIONS 3

ny -p(t)y- q(t)y

The case of n being even is considered in the following theorem.

THEOREM 2.2. If (I) p(t) < 0, t >0, (2) q(t) < 0, t > 0 and (3) n

is even, then y(t) < O, for all t > 0.

PROOF. Since q(t) < 0 and n is even, we have

+ p(t)y + q(t)yn < + p(t)y,

therefore

0 < + p(t)y

and by Bellman and Kalaba [13, p. 67], y(t) < 0 for all t > 0.

CASE O p(C) < 0 AND q(=) > O.

THEOREM 3.1. If (I) p(t) < 0, t > 0, (2) q(t) > 0, t > 0 and (3) n

is even, then y(t) > 0, for all t > 0.

PROOF. METHOD i. If in (I.i), we let y(t) -z(t), then the equation be-

comes

-E- p(t)z + q(t)(-l)nzn-- 0,

since n is even, . + p(t)z q(t)zn 0

and by Theorem 2.2, z(t) < 0 for all t > 0. Therefore y(t) > 0 for all t > 0.

METHOD 2. Since q(t) > 0 and n is even, we have

y + p(t)y < y + p(t)y + q(t)y

therefore

+ p(t)y < 0

and by Bellman and Kalaba [13, p. 67], y(t) > 0 for all t > 0.

Page 4: Behavior of SON Differential Equations

4 RINA LING

4. CASE OF p(t) > 0 AND q(t) < 0.

If n is odd, the following two theorems on the number of zeroes of the

solution can be obtained.

THEOREM 4.1. If (i) p(t) > 0, a < t < b, (2) q(t) < 0, a < t < b and

(3) n is odd, then a necessary condition for y to have two zeroes on (a, b] is

that

b

p(t)dt >

a

4

PROOF. Equation (i.i) can be written in the form

+ (p(t) + q(t)yn-l)y 0.

Let z(t) be real-valued and satisfy the linear equation

+ p(t)z 0.

Since q(t) < 0 and (n- i) is even,

p(t) + q(t)yn-I < p(t)

and by Hrtman [12], z(t) has at least two zeroes on (a, b) if y(t) has two

zeroes on (a, b]. By Lyapunov Theorem in Hartman [12, p. 346], a necessary

condition for z(t) to have two zeroes on [a, b] is that

b

p(t)dt > 4

THEOREM 4.2. If (i) p(t) > 0, 0 < t < T, (2) q(t) < 0, 0 < t < T,

(3) n is odd, (4) y(t) has N zeroes on (0,T], then

T

N < (T p(t)dt) + i.

0

Page 5: Behavior of SON Differential Equations

NONLINEAR DIFFTIAL EQUATIONS 5

PROOF. As in the proof of Theorem 4.1, it can be shown that if z(t) has

M zeroes on (0,T), then N <_ M. But by Hartman [12, p. 346-347 ],

T

M < (T p(t)dt + i

0

and the conclusion follows.

In the next two theorems, n is assumed to be even.

THEOREM 4.3. If (I) p(t) > 0, t >_ 0, (2) q(t) < 0, t >_ 0 and (3) n

is even, then y(t) < 0, for t > 0.

PROOF. Since q(t) < 0 and n is even,

+ p(t)y + q(t)yn < + p(t)y,

therefore,

0 < ; + p(t)y

and by Bellman and Kalaba [13, p. 67], y(t) <_ 0, for all t > 0.

If in addition to the hypotheses of Theorem 4.3, we assume that (0) > 0,

then y is negative and monotonic increasing.

THEOREM 4.4. If (i) (0) > 0, (2) p(t) > 0, t > 0, (3) q(t) < 0,

t > 0 and (4) n is even, then y is montonic increasing.

PROOF. Integration of (i.i) from 0 to t leads to

(t) (0)+ I p(s)yds + I q(s)yn ds 0.

0 0

Since p > 0 and y < 0 by Theorem 4.3, q < 0 and n is even,

(t) (0) " r(t) (0) + I p(s)yds + I q(s)yn ds,

0 0

Page 6: Behavior of SON Differential Equations

6 RINA LING

therefore,

(t) (0) > O,

(t) > (o) > o

and so y is monotonic increasing.

5. CASZ 0 p(=) > 0 AD q(t) > 0.

For p(t) > 0, q(t) > 0 and n odd, the following theorems on the oscillatory

behavior and boundedness of the solutions can be obtained.

THEOREM 5.1. (i) If p(t) > 0, t > 0, (2) q(t) > 0, t > 0, (3) n is

odd and (4) z(t) is a real-valued solution to + p(t) z 0, then y(t)

oscillates more rapidly than z(t), for t > 0.

PROOF. Equatlon (I.i) can be written in the form

+ (p(t) + q(t)yn-1)y O.

Let z(t) be a real-valued solution to

E + p(t)z 0

which has been widely discussed. Since q(t) > 0 and (n i) is even,

p(t) < p(t) + q(t)yn-1

and the conclusion follows from comparison theorems in Hartman and Sanchez [12,

14].

THEOREM 5.2. If (i) p(t) > 0, (t) > 0, t >_ 0, (2) q(t) > 0, (t) > 0,

t > 0, (3) n is odd and (4) y has successive extrema at tl, t2, tI< t2,

then lY(t2) < lY(tl)I. (The amplitudes of oscillations do not grow.)

PROOF. The proof is by contradiction, so assume that lY(tl) < [Y(t2)Multiplication of (i.i) by leads to

+ p(t) + q(t)yn 0.

Page 7: Behavior of SON Differential Equations

NONLINEAR DIFFERENTIAL EQUATIONS 7

Integrating (5.1) from t tI to t t2, we get

t2

t2

I P(t) dt + I q(t)Yn dt 0.

tI tI

Since

t2 t2

p(t) at ;g (p(t2)y2(t2) P(ti)y2(ti) (t)ytI t1

2 dt)

and

t2

q(t)yn dt

tI

t2i yn+1 yn+ [ yn+

n + i (q(t2) (t2) q(tI) (tI) (t) dr),

tI

(5.2) becomes

P(t2)Y2(t2) P(tl)Y2(tl) + 2 yn+n + 1 q(t2) (t2) 2 )yn+l

n + I q(tl (tl)

t2 t2

I (t)y2 dt +n +12 I (t)yn+1 dt

tI tI

< y2(t2)(P(t2) P(tl)) + 2 n+ln + i

y (t2)(q(t2) q(tl))’

therefore

2 (yn+ yn+l (t)P(tl)(y2(t2) y2(tl)) <n + i q(tl) (tl) 2

< 0, since q(tl) > 0 and (n + i) is even,

which is a contradiction since the left hand side is positive.

THEOREM 5.3. If (i) p(t) > i, (t) < 0, t > 0, (2) q(t) > 0, (t) < 0

and (3) n is odd, then y(t) is bounded for all t.

Page 8: Behavior of SON Differential Equations

8 RINA LING

PROOF. Multiplication of (1.1) by and integration of the resulting

equation from 0 to t lead to

t

#2(t) + p(t) y2(t) I (s)y2 ds +0

where C is a constant,

therefore

t2 yn+l 2 I (s)yn+lds C,n+ q() ()

n+l0

t2 yn- I 2y2(t)(p(t) + q(t) (t)) (s)y ds

0

t

0

Since p > i, q > 0, n is odd, < 0 and < 0, it follows that

y2(t) < C, for all t

and so y is bounded.

THEOREM 5.4. If (I) p(t) > 0, (t) > 0, t > 0, (2) q(t) > 0,

(t) < 0, t > 0, and (3) n is odd, then y(t) is bounded for all t.

PROOF. As in Theorem 5.3, equation (i.I) is multiplied by and the

resulting equation integrated from 0 to t, giving,

2)yn+2(t) + P(t)y2(t) + n--- q(t (t)

t2 In+l

ds

0

=C+ (s)y2 ds.

0

Since q > 0, (n + i) is even and <_ 0,

t

p(t)y2(t) < C + I (s)y2 ds

0

so

t

p(t)y2(t) _< [C[ + I P(s)Y2 (S)p(s) ds

0

Page 9: Behavior of SON Differential Equations

NONLINEAR DIFFERENTIAL EQUATIONS 9

and by Gronwall’ s inequality, Hartman [12, p. 24

t

p(t)y2(t) _< ICI exp I p(s)(s) ds

0

IC p(t)p(0)

therefore

y2(t) <p for all t.

In the next theorem, (i.i) is assumed to have constant coefficients P0and q0"

THEOREM 5.5. If (i) P0 > 0, (2) q0 > 0 and (3) n is odd, then y is

oscillatory.

PROOF. The equation is

9 + pOy + q0yn 0.

By linearization in McLachlan [12, p. 106],

where

9+ay=0,

2i I Ana (P0 A sin e + q0 sin

n e) sine de,

0

where A is a constant.

Since P0 > 0, q0 > 0 and n is odd, the integrand in

2i I An- inn+!a --w (P0 sin2 e + q0 s e)de

0

is positive, so a > 0.

Page 10: Behavior of SON Differential Equations

i0 RINA LING

By Theorem 5.4, y is bounded. The fact that

ldt=m

0

and y is bounded imply that y is oscillatory, see Hartman [12, p. 354].

In the next two theorems, n is assumed to be even.

THEOREM 5.6. If (i) p(t) > 0, t > 0, (2) q(t) > 0, t > 0 and (3) n

is even, then y(t) > 0, for t > 0.

PROOF. The equation is

+ p(t)y + q(t)yn 0.

Let y(t) -z(t), then

-. p(t)z + (-l)nq(t) zn 0.

Since n is even,

+ p(t)z q(t)zn 0

and by Theorem 4.3, z < 0. Therefore y(t) > 0 for t > 0.

If in addition to the hypotheses of Theorem 5.6, we assume that (0) < 0,

then y is positive and monotonic decreasing.

THEORE .7. Tf () (0) ! 0, (2) p() > 0, >_ 0, (3) q() > 0,

and (4) n is even, then y is monotonic decreasing.

PROOF. Integration of (i.I) from 0 to t leads to

t t

(t)- (0)+ f p(s)y ds + I q(s)tn ds 0"

0 0

Since p > 0 and y > 0 by Theorem 5.6, q > 0 and n is even,

Page 11: Behavior of SON Differential Equations

NONLINEAR DIFFERENTIAL EQUATIONS 11

(t) (0) < O,

so (t) < ;(o) < o

and y is monotonic decreasing.

REFERENCES

i. Ames, W. F. Nonlinear. Ordln.ary Differential Equatl.o.ns in Tra.nsp.ortProcesses., Academic Press, New York and London, 196’8.

2. McLachlan, N. W. Ordinary Non-Linear Differential Equation.s. in Engin.ee.ringand Physlcal Sciences, 2nd ed., Oxford University Press, London and NewYork, 1958.

3. Struble, R. A. Nonlinear Differential Equations, McGraw-Hill Company, NewYork and London, 1962.

4. Canosa, J. and J. Cole. Asymptotic Behavior of Certain Nonlinear Boundary-Value Problems, J. Math. Phys. 9 (1968) 1915-1921.

5. Canosa, J. Expansion Method for Nonlinear Boundary-Value Problems, J. Math.Phys. 8 (1967) 2180-2183.

6. Skidmore, A. On the Differentlal Equation y" + p(x)y f(x), J. Math. Anal.App1. 43 (1973) 46-55.

7. Abramovich, S. On the Behavior of the Solutions of y" + p(x)y f(xi, J.Math. Anal. Appl. 52 (1975) 465-470.

8. Rankin, S. M. Oscillation Theorems for Second-Order Nonhomogeneous LinearDifferential Equations, J. Math. Anal. Appl. 53 (1976) 550-553.

9. Grimmer, R. C. and W. T. Patula. Nonoscillatory Solutions of Forced Second-Order Linear Equations, J. Math. Anal. Appl. 56 (1976) 452-459.

i0. Chen, L. S. Some Oscillation Theorem for Differential Equations withFunctional Arguments, J. Math. Anal. Appl. 58 (1977) 83-87.

ii. Chen, M. P., C. C. Yeh and C. S. Yu. Asymptotic Behavior of NonoscillatorySolutions of Nonlinear Differential Equations with Retarded Arguments,J. Math. Anal. Appl. 59 (1977) 211-215.

12. Hartman, P. Ordinary Differential Equations, John Wiley and Sons, New Yorkand London, 1964.

Page 12: Behavior of SON Differential Equations

12 RINA LING

13. Bellman, R. E. and R. E. Kalaba. uasilinearizatlon and Nonlinear Boundary-Value Problems, American Elsevler Publishing Company, New York, 1965.

14. Sanchez, D. A. Ordinary Differential Equations and Stability Theory,W. H. Freeman and Company, San Francisco and London, 1968.

KEF WORDS AND PHRASES. Nonlin differential equation, osry andymptotie behavior of solons, boundedn and onotonity of olons.

AMS(MOS) SUBJECT CLASSIFICATIONS (1970). 34C10, 34C15, 34D05.